For a real number $q>1$ and a positive integer $m$, let
$$
Y_m(q):=\left\{\sum_{i=0}^n\epsilon_i q^i:\; \epsilon_i\in \{0, \pm 1,\ldots, \pm m\},\; n=0, 1,\ldots \right\}.
$$
In this paper, we show that $Y_m(q)$ is dense in $\mathbb R$ if and only if $q < m + 1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdős, Joó and Komornik [8].